# OS-V: 0531 Laminated Shell Strength Analysis Mechanical Load 1

This problem analyzes the strength of laminated composite shells when subjected to a uniform longitudinal load per unit length.

The model and boundary conditions are described by Hopkins (2005). The resulting ply failure indices are compared against analytical solutions from classical lamination theory (CLT). The results indicate good agreement between OptiStruct and CLT.

## Benchmark Model

800 mesh elements of CQUAD4 element type were used in this study. The model is fixed at the point A using a SPC card and a uniform longitudinal load per unit length (Nx) of 1500 N/m is applied along edges A-D and B-C using FORCE card.

The material properties are:
Property
Value
Longitudinal Young’s Modulus, El (GPa)
207.0
Transverse Young’s Modulus, Et (GPa)
7.6
Longitudinal Shear Modulus, Glt (GPa)
5.0
Major Poisson’s ratio, $\upsilon$ 12
0.3
Longitudinal Tensile Strength, $\sigma$ lt (MPa)
500.0
Longitudinal Compressive Strength, $\sigma$ lc (MPa)
350.0
Transverse Tensile Strength, $\sigma$ tt (MPa)
5.0
Transverse Compressive Strength, $\sigma$ tc (MPa)
75.0
In-plane shear strength, $\tau$ lt (MPa)
35.0

Ply Orientation (°) Thickness ( $\mu$ m)
1 90.0 0.05
2 -45.0 0.05
3 45.0 0.05
4 0.0 0.05
The geometry of the composite laminate:
Dimension
Value
Length (m)
0.2
0.1

## Results

Midplane Strains Theory OptiStruct Result
$\text{ε}$ x 3.176 x 10-4 3.176 x 10-4
$\text{ε}$ y -1.447 x 10-4 -1.447 x 10-4
$\text{ε}$ xy 1.108 x 10-4 1.108 x 10-4

Failure Criteria Ply 1 Ply 2 Ply 3 Ply 4
Theory OptiStruct Result Theory OptiStruct Result Theory OptiStruct Result Theory OptiStruct Result
Tsai-Wu 0.88402 0.8841 0.37308 0.3731 0.01990 0.01991 -0.34309 -0.343
Hill 0.77952 0.77960 0.16323 0.16330 0.00435 0.00435 0.00136 0.00136
Hoffman 0.88110 0.88140 0.37630 0.37610 0.02004 0.02000 -0.34534 -0.34510

Reserve Factor Ply 1 Ply 2 Ply 3 Ply 4
Theory OptiStruct Result Theory OptiStruct Result Theory OptiStruct Result Theory OptiStruct Result
Tsai-Wu 1.1223 1.122 2.53671 2.537 14.304 14.3 31.879 31.89
Hill 1.1325 1.133 2.4748 2.475 15.157 15.16 27.124 27.12
Hoffman 1.1259 1.126 2.4944 2.494 14.101 14.1 37.869 37.88

## Model Files

The model files used in this problem include:
• lssam1_tsai.fem
• lssam1_hill.fem
• lssam1_hoff.fem

## Reference

NAFEMS R0092 - Benchmarks for membrane and bending analysis of laminated shells. Part 1, Stiffness matrix and thermal characteristics

NAFEMS R0093 - Benchmarks for membrane and bending analysis of laminated shells. Part 2, Strength analysis